Difference between revisions of "1973 USAMO Problems/Problem 4"

Latest revision as of 07:18, 22 April 2024

Problem

Determine all the roots, real or complex, of the system of simultaneous equations

$x+y+z=3$ ,

$x^2+y^2+z^2=3$ ,

$x^3+y^3+z^3=3$ .

Solution 1

Let $x$ , $y$ , and $z$ be the roots of the cubic polynomial $t^3+at^2+bt+c$ . Let $S_1=x+y+z=3$ , $S_2=x^2+y^2+z^2=3$ , and $S_3=x^3+y^3+z^3=3$ . From this, $S_1+a=0$ , $S_2+aS_1+2b=0$ , and $S_3+aS_2+bS_1+3c=0$ . Solving each of these, $a=-3$ , $b=3$ , and $c=-1$ . Thus $x$ , $y$ , and $z$ are the roots of the polynomial $t^3-3t^2+3t-1=(t-1)^3$ . Thus $x=y=z=1$ , and there are no other solutions.

Solution 2

Let $P(t)=t^3-at^2+bt-c$ have roots x, y, and z. Then $0=P(x)+P(y)+P(z)=3-3a+3b-3c$ using our system of equations, so $P(1)=0$ . Thus, at least one of x, y, and z is equal to 1; without loss of generality, let $x=1$ . Then we can use the system of equations to find that $y=z=1$ as well, and so $\boxed{(1,1,1)}$ is the only solution to the system of equations.

Solution 3

Let $a=x-1,$ $b=y-1$ and $c=z-1.$ Then $a+b+c=0,$ $a^2+b^2+c^2=0,$ $a^3+b^3+c^3=0.$ We have $\begin{align*} 0&=(a+b+c)^3\\ &=(a^3+b^3+c^3)+3a^2(b+c)+3b^2(a+c)+3c^2(a+b)+6abc\\ &=0-3a^3-3b^3-3c^3+6abc\\ &=6abc. \end{align*}$ Then one of $a, b$ and $c$ has to be 0, and easy to prove the other two are also 0. So $\boxed{(1,1,1)}$ is the only solution to the system of equations.

J.Z.

Solution 4

We are going to use Intermediate Algebra Techniques to solve this equation.

Let's start with the first one: $x+y+z=3$ . This will be referred as the FIRST equation.

We are going to use the first equation to relate to the SECOND one ( $x^2+y^2+z^2=3$ ) and the THIRD one ( $x^3+y^3+z^3=3)$ .

Squaring this equation: $x^2+y^2+z^2+2xy+2yz+2xz=9$

Subtracting this equation from the 2nd equation in the problem, we have $2xy+2yz+2xz=6$ , so $xy+xz+yz=3$ .

Now we try the same idea with the cubed terms. Cube the first equation:

$x^3+y^3+z^3+3xy^2+3xz^2+3yx^2+3yz^2+3zx^2+3zy^2=27$ . Plug in $x^3+y^3+z^3=3$ and factor partially:

$3+3(x^2(y+z)+y^2(x+z)+z^2(x+y))+6xyz=27$ .

Now here is the key step. Note that $z=3-x-y, y=3-x-z, x=3-y-z$ . So we are going to substitute $y+z, x+z, x+y$ for each of the expressions and we get: $-x^3+3x^2-y^3+3y^2-z^3+3z^2=8-2xyz$ (I rearranged it a bit).

Resubstituting in the second and third equation: $-3+3(3)=8-2xyz$ . So $xyz=1$ .

So now we have three equations for the elementary symmetric sums of $x,y,z$ :

Equation 4: $x+y+z=3$ (this is also equation 1)

Equation 5: $xy+yz+xz=3$

Equation 6: $xyz=1$ .

If we call the solutions of $t^3-3t^2+3t-1=0$ (Equation 7) $a,b,c$ , then $x,y,z$ are the three roots $a,b,c$ but in some order. Notice that Equation 7 can be factored as $(t-1)^3=0$ , which means that $t=1$ . Therefore $(x,y,z)$ are permutations of $(1,1,1)$ in some order, which can be only $(1,1,1)$ . (This step uses Vieta's formulas)

Therefore, the only solution is $\boxed{(x,y,z)=(1,1,1)}$ .

~hastapasta

Solution 5

From the question

Equation 1 : $x+y+z=3$

Equation 2 : $x^2+y^2+z^2=3$

Equation 3 : $x^3+y^3+z^3=3$

From Eq(1) and Eq(2),

Equation 4 : $(x+y+z)^2=9 \implies x^2+y^2+z^2+2\sum xy = 9 \implies \sum xy = 3$

From Eq(1), Eq(3), and the fact that $(x+y+z)^3=x^3+y^3+z^3+3\prod(x+y)$ ,

Equation 5 : $(x+y+z)^3=x^3+y^3+z^3+3\prod(x+y)= 27 \implies \prod(x+y)=8$

As from Eq(1), $\ x+y=3-z$ , $\ y+z=3-x$ , $\ z+x = 3- y$

Equation 6 : $\prod(3-x) =8$

Now let there be a cubic function $f(t)$ with the roots $x,y,z$ so from the relation between roots and coefficients,

Equation 7 : $f(t)= t^3-3t^2+3t-d$ (As we don't know the product let the constant be $d$ )

As $f(t) = (t-x)(t-y)(t-z)$ then at $t=3$ ,

Equation 8 : $f(3)=9-d=(3-x)(3-y)(3-z)$

And from Eq(6),

Equation 9 : $d= -1$

So finally,

Equation 10 : $f(t)=t^3-3t^2+3t-1=(t-1)^{3}$

So, $\ (x,y,z)=(1,1,1)$

~Sparrow_1827

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

@@ Line 5: / Line 5: @@
 <math>x^3+y^3+z^3=3</math>.</center>
-==Solution==
+==Solution 1==
-Let <math>x</math>, <math>y</math>, and <math>z</math> be the roots of the cubic <math>x^3+ax^2+bx+c</math>. Let <math>S_1=x+y+z=3</math>, <math>S_2=x^2+y^2+z^2=3</math>, and <math>S_3=x^3+y^3+z^3=3</math>. From this, <math>S_1+a=0</math>, <math>S_2+aS_1+b=0</math>, and <math>S_3+aS_2+bS_1+c=0</math>. Solving each of these, <math>a=-3</math>, <math>b=3</math>, and <math>c=-1</math>. Thus <math>x</math>, <math>y</math>, and <math>z</math> are the roots of the polynomial <math>x^3-3x^2+3x-1=(x-1)^3</math>. Thus <math>x+y+z=1</math>, and there are no other solutions.
+Let <math>x</math>, <math>y</math>, and <math>z</math> be the [[root]]s of the [[cubic polynomial]] <math>t^3+at^2+bt+c</math>. Let <math>S_1=x+y+z=3</math>, <math>S_2=x^2+y^2+z^2=3</math>, and <math>S_3=x^3+y^3+z^3=3</math>. From this, <math>S_1+a=0</math>, <math>S_2+aS_1+2b=0</math>, and <math>S_3+aS_2+bS_1+3c=0</math>. Solving each of these, <math>a=-3</math>, <math>b=3</math>, and <math>c=-1</math>. Thus <math>x</math>, <math>y</math>, and <math>z</math> are the roots of the polynomial <math>t^3-3t^2+3t-1=(t-1)^3</math>. Thus <math>x=y=z=1</math>, and there are no other solutions.
-==See also==
+==Solution 2==
+Let <math>P(t)=t^3-at^2+bt-c</math> have roots x, y, and z. Then <cmath>0=P(x)+P(y)+P(z)=3-3a+3b-3c</cmath> using our system of equations, so <math>P(1)=0</math>. Thus, at least one of x, y, and z is equal to 1; without loss of generality, let <math>x=1</math>. Then we can use the system of equations to find that <math>y=z=1</math> as well, and so <math>\boxed{(1,1,1)}</math> is the only solution to the system of equations.
+==Solution 3==
+Let <math>a=x-1,</math> <math>b=y-1</math> and <math>c=z-1.</math> Then
+<cmath>a+b+c=0,</cmath>
+<cmath>a^2+b^2+c^2=0,</cmath>
+<cmath>a^3+b^3+c^3=0.</cmath>
+We have
+<cmath>\begin{align*}
+&=(a+b+c)^3\\
+&=(a^3+b^3+c^3)+3a^2(b+c)+3b^2(a+c)+3c^2(a+b)+6abc\\
+&=0-3a^3-3b^3-3c^3+6abc\\
+&=6abc.
+\end{align*}</cmath>
+Then one of <math>a, b</math> and <math>c</math> has to be 0, and easy to prove the other two are also 0. So <math>\boxed{(1,1,1)}</math> is the only solution to the system of equations.
+J.Z.
+== Solution 4 ==
+We are going to use Intermediate Algebra Techniques to solve this equation.
+Let's start with the first one: <math>x+y+z=3</math>. This will be referred as the FIRST equation.
+We are going to use the first equation to relate to the SECOND one (<math>x^2+y^2+z^2=3</math>) and the THIRD one (<math>x^3+y^3+z^3=3)</math>.
+Squaring this equation: <math>x^2+y^2+z^2+2xy+2yz+2xz=9</math>
+Subtracting this equation from the 2nd equation in the problem, we have <math>2xy+2yz+2xz=6</math>, so <math>xy+xz+yz=3</math>.
+Now we try the same idea with the cubed terms. Cube the first equation:
+<math>x^3+y^3+z^3+3xy^2+3xz^2+3yx^2+3yz^2+3zx^2+3zy^2=27</math>. Plug in <math>x^3+y^3+z^3=3</math> and factor partially:
+<math>3+3(x^2(y+z)+y^2(x+z)+z^2(x+y))+6xyz=27</math>.
+Now here is the key step. Note that <math>z=3-x-y, y=3-x-z, x=3-y-z</math>. So we are going to substitute <math>y+z, x+z, x+y</math> for each of the expressions and we get: <math>-x^3+3x^2-y^3+3y^2-z^3+3z^2=8-2xyz</math> (I rearranged it a bit).
+Resubstituting in the second and third equation: <math>-3+3(3)=8-2xyz</math>. So <math>xyz=1</math>.
+So now we have three equations for the elementary symmetric sums of <math>x,y,z</math>:
+Equation 4: <math>x+y+z=3</math> (this is also equation 1)
+Equation 5: <math>xy+yz+xz=3</math>
+Equation 6: <math>xyz=1</math>.
+If we call the solutions of <math>t^3-3t^2+3t-1=0</math> (Equation 7) <math>a,b,c</math>, then <math>x,y,z</math> are the three roots <math>a,b,c</math> but in some order. Notice that Equation 7 can be factored as <math>(t-1)^3=0</math>, which means that <math>t=1</math>. Therefore <math>(x,y,z)</math> are permutations of <math>(1,1,1)</math> in some order, which can be only <math>(1,1,1)</math>. (This step uses Vieta's formulas)
+Therefore, the only solution is <math>\boxed{(x,y,z)=(1,1,1)}</math>.
+~hastapasta
+== Solution 5 ==
+From the question
+Equation 1 : <math>x+y+z=3</math>
+Equation 2 : <math>x^2+y^2+z^2=3</math>
+Equation 3 : <math>x^3+y^3+z^3=3</math>
+From Eq(1) and Eq(2),
+Equation 4 : <math>(x+y+z)^2=9 \implies x^2+y^2+z^2+2\sum xy = 9 \implies \sum xy = 3</math>
+From Eq(1), Eq(3), and the fact that <math>(x+y+z)^3=x^3+y^3+z^3+3\prod(x+y)</math>,
+Equation 5 : <math>(x+y+z)^3=x^3+y^3+z^3+3\prod(x+y)= 27 \implies \prod(x+y)=8</math>
+As from Eq(1),  <math>\ x+y=3-z</math>,<math>\ y+z=3-x</math>,  <math>\ z+x = 3- y</math>
+Equation 6 : <math>\prod(3-x) =8</math>
+Now let there be a cubic function <math>f(t)</math> with the roots <math>x,y,z</math> so from the relation between roots and coefficients,
+Equation 7 : <math>f(t)= t^3-3t^2+3t-d</math>   (As we don't know the product let the constant be <math>d</math>)
+As <math>f(t) = (t-x)(t-y)(t-z)</math> then at <math>t=3</math>,
+Equation 8 : <math>f(3)=9-d=(3-x)(3-y)(3-z)</math>
+And from Eq(6),
+Equation 9 : <math>d= -1</math>
+So finally,
+Equation 10 : <math>f(t)=t^3-3t^2+3t-1=(t-1)^{3}</math>
+So, <math>\ (x,y,z)=(1,1,1)</math>
+~Sparrow_1827
+{{alternate solutions}}
+==See Also==
 [[Newton's Sums]]
-{{USAMO oldbox|year=1973|num-b=3|num-a=5}}
+{{USAMO box|year=1973|num-b=3|num-a=5}}
+{{MAA Notice}}
 [[Category:Olympiad Algebra Problems]]

1973 USAMO (Problems • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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